In this paper, we consider the boundary value problem for the following quasilinear Schrödinger equation: \begin{align*} & \int_\Omega\sum_{ij=1}^N a_{ij}(x, u)D_i uD_j\varphi dx+\frac{1}{2}\int_\Omega\sum_{ij=1}^ND_sa_{ij}(x, u)D_iuD_ju\varphi dx\\ & \notag \quad +\displaystyle\int_{\partial\Omega}g(x)u\varphi dx=\int_{\partial\Omega}f(x, u)\varphi dx, \forall \varphi\in C^\infty(\bar\Omega), \tag*{(P)} \end{align*} where $\Omega\subset\mathbb{R}^N (N\geq 3)$ is a smooth bounded domain, $D_i=\frac{\partial}{\partial x_i},$ $ D_sa_{ij}(x, s)=\frac{\partial}{\partial s}a_{ij}(x, s). $ These kind of equations include the so-called Modified Nonlinear Schrödinger Equation (MNLS). By using a perturbation method, we prove the existence of infinitely many solutions for the problem (P).

We study the asymptotic behavior of blow-up solutions of the heat equation with nonlinear boundary conditions. In particular, we classify the asymptotic behavior of blow-up solutions and investigate the spacial singularity of their blow-up profiles.

By a Lyapunov-Schmidt reduction argument, for a type of Grushin critical problem with cylindrical symmetry, we prove that it has infinitely many positive solutions with cylindrical symmetry, whose energy can be made arbitrarily large.

We demonstrate the existence of at least one positive solution to \begin{equation} \begin{split} -y''(t)& =\lambda f(t,y(t))\text{, }t\in(0,1)\\ y(0)& =H(\varphi(y))\text{, }y(1)=0,\notag \end{split} \end{equation} where $H : \mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and $\varphi : \mathcal{C}([0,1])\rightarrow\mathbb{R}$ is a linear functional so that the boundary condition at $t=0$ may be both nonlocal and nonlinear. Since the continuous function $f : [0,1]\times\mathbb{R}\rightarrow\mathbb{R}$ may assume negative values, our results apply to semipositone problems. The classical Leray-Schauder degree is utilized to derive the existence result, which we obtain in the case where $\lambda$ is small and which permits $f$ to be negative on its entire domain. The result is illustrated by an example.

This paper is concerned with the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation whose coefficient is integrable in time. We show the unique existence of global solutions for small data in some class and that the solution has the same asymptotic behavior of a function obtained by a transformation of time variable from a solution of the free wave equation with an appropriate wave speed. Conversely, we show that there exists a solution of the Kirchhoff equation which has the same asymptotic behavior of a function obtained by a transformation of the time variable from the solution of the Cauchy problem of the free wave equation with an appropriate wave speed.